Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}3x+2y &= -9 \\ -4x-y &= 5\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $-4x = y+5$ Divide both sides by $-4$ to isolate $x$ $x = {-\dfrac{1}{4}y - \dfrac{5}{4}}$ Substitute this expression for $x$ in the first equation. $3({-\dfrac{1}{4}y - \dfrac{5}{4}}) + 2y = -9$ $-\dfrac{3}{4}y - \dfrac{15}{4} + 2y = -9$ Simplify by combining terms, then solve for $y$ $\dfrac{5}{4}y - \dfrac{15}{4} = -9$ $\dfrac{5}{4}y = -\dfrac{21}{4}$ $y = -\dfrac{21}{5}$ Substitute $-\dfrac{21}{5}$ for $y$ in the top equation. $3x+2( -\dfrac{21}{5}) = -9$ $3x-\dfrac{42}{5} = -9$ $3x = -\dfrac{3}{5}$ $x = -\dfrac{1}{5}$ The solution is $\enspace x = -\dfrac{1}{5}, \enspace y = -\dfrac{21}{5}$.